JournalsjemsVol. 15, No. 5pp. 1597–1628

Control for Schrödinger operators on 2-tori: rough potentials

  • Jean Bourgain

    Institute for Advanced Study, Princeton, United States
  • Nicolas Burq

    Université Paris-Sud, Orsay, France
  • Maciej Zworski

    University of California, Berkeley, USA
Control for Schrödinger operators on 2-tori: rough potentials cover
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Abstract

For the Schr\"odinger equation, (it+Δ)u=0( i \partial_t + \Delta ) u = 0 on a torus, an arbitrary non-empty open set Ω\Omega provides control and observability of the solution: ut=0L2(\T2)KTuL2([0,T]×Ω)\| u |_{ t = 0 } \|_{ L^2 ( \T^2 ) } \leq K_T \| u \|_{ L^2 ( [0,T] \times \Omega )} . We show that the same result remains true for (it+ΔV)u=0( i \partial_t + \Delta - V ) u = 0 where VL2(\T2)V \in L^2 ( \T^2 ) , and \T2\T^2 is a (rational or irrational) torus. That extends the results of \cite{AM}, and \cite{BZ4} where the observability was proved for VC(\T2)V \in C ( \T^2) and conjectured for VL(\T2)V \in L^\infty ( \T^2 ) . The higher dimensional generalization remains open for VL(\Tn)V \in L^\infty ( \T^n )

Cite this article

Jean Bourgain, Nicolas Burq, Maciej Zworski, Control for Schrödinger operators on 2-tori: rough potentials. J. Eur. Math. Soc. 15 (2013), no. 5, pp. 1597–1628

DOI 10.4171/JEMS/399